On Convergence of the Immersed Boundary Method for Elliptic Interface Problems

On convergence of the immersed boundary method for elliptic interface problems

Zhilin Li∗

January 26, 2012

Abstract

Peskin’s Immersed Boundary (IB) method is one of the most popular numerical methods for many years and has been applied to problems in mathematical biology, fluid mechanics, material sciences, and many other areas. Peskin’s IB method is associated with discrete delta functions. It is believed that the IB method is first order accurate in the L∞ norm. But almost no rigorous proof could be found in the literature until recently [14] in which the author showed that the velocity is indeed first order accurate for the Stokes equations with a periodic boundary condition. In this paper, we show first order convergence with a log h factor of the IB method for elliptic interface problems essentially without the boundary condition restrictions. The results should be applicable to the IB method for many different situations involving elliptic solvers for Stokes and Navier-Stokes equations.

keywords: Immersed Boundary (IB) method, Dirac delta function, convergence of IB method, discrete Green function, discrete Green’s formula. AMS Subject Classiﬁcation 2000 65M12, 65M20.

1 Introduction

Since its invention in 1970’s, the Immersed Boundary (IB) method [15] has been applied almost everywhere in mathematics, engineering, biology, ﬂuid mechanics, and many many more areas, see for example, [16] for a review and references therein. The IB method is not only a mathematical modeling tool in which a complicated boundary condition can be treated as a source distribution but also a numerical method in which a discrete delta function is used. The IB method is robust, simple, and has been applied to many problems.

∗Center for Research in Scientiﬁc Computation (CRSC) and Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA

1 It is widely believed that Peskin’s IB method is only first order accurate in the L∞ norm. How- ever, there was almost no rigorous proof in the literature until recently [14], in which the author has proved the first order accuracy of the IB method for the Stokes equations with a periodic boundary condition. The proof is based on some known inequalities between the fundamental solution and the discrete Green function with a periodic boundary condition for Stokes equations. In [4], the author showed that the pressure obtained from IB method has O(h1/2) order of convergence in the L2 norm for a 1D model. In [19, 20], the authors designed some level set methods based on discrete delta functions. With suitable quadrature formulas in the integral form using the Green functions, the authors show that their approach can get expected accuracy. However, there are few theoretical proofs on the IB method for elliptic interface problems with general boundary conditions. This is the main motivation of this paper. One difficult is that there is little known estimates between the fundamental solution and the discrete Green function with Dirichlet or other boundary conditions on rectangular domains. Compared with the case of periodic boundary conditions where there are existing estimates between the discrete Green function and the continuous one [8], there is almost none for Dirichlet and other boundary conditions. The main goal of this paper is to provide a convergence proof for the IB method for elliptic interface problems with Dirichlet boundary conditions. We will show that with commonly used discrete delta functions that satisfy the zeroth moment condition and first order interpolation property, the IB method is indeed first order convergent in the L∞ norm with a log h factor. The key in our proof is to establish a connection between the discrete Green function and the continuous one. Our proof is essentially independent of the boundary conditions and it is valid in 1D, 2D, and 3D cases. The result should be applicable for many IB methods involving Stokes and Navier-Stokes solvers.

2 Proof of the convergence of the IB method in 1D

We will give a proof for the 1D model,

u00 = c δ(x α) 0 < x < 1, 0 < α < 1, u(0) = u(1) = 0, (1) − ﬁrst in this section. Note that the analytic solution to the equation above is

cx (1 α) if x α, u(x) = − − ≤ (2) ( cα (1 x) otherwise. − −

Given a uniform Cartesian grid x = ih, i = 0, 1, , n, h = 1/n, the IB method leads to the i · · · following system of linear equations,

Ui 1 2Ui + Ui+1 − − = cδ (x α) , i = 1, 2, , n 1, (3) h2 h i − · · · − where U is the finite difference approximation of the solution u(x ), and δ (x α) is a discrete delta i i h i − function applied to the grid point xi. In the matrix-vector form, the above finite difference equations can be written as AhU = F, where Ah is the tri-diagonal matrix formed by the discrete Laplacian.

2 It is well known that A is a symmetric positive deﬁnite matrix (SPD) and diagonally dominant. − h Note that, a discrete delta function has a compact support in the neighborhood of interface, that is, δ (x) = 0 only if x W h, where W is a constant. Commonly used discrete delta functions include h 6 | | ≤ the hat discrete delta function (δhat(x) with W = 1):

(h x )/h2, if x < h, hat − | | | | δh (x) = (4) ( 0, if x h, | | ≥ and Peskin’s original discrete cosine delta function (δcosine(x) with W = 2) 1 (1 + cos (πx/2h)) , if x < 2h, δ (x)cosine = 4h | | (5) h  0, if x 2h.  | | ≥ see for example, [13]. Note that, when we use the hat delta function, the result is the same as that of the IIM for the simple model. The solution to the finite difference equations is the same as true solution if there are no round-off errors, that is

u(xi 1) 2u(xi) + u(xi+1) hat − − = cδ (x α) , i = 1, 2, , n 1, (6) h2 h i − · · · − see for example, [3, 13]. But this is not the case for other discrete delta functions. We deﬁne the error vector as E = E , where E = u(x ) U . The local truncation error is { i} i i − i deﬁned as T = T , { i}

u(xi 1) 2u(xi) + u(xi+1) T = − − cδ (x α) . (7) i h2 − h i −

With the definition, we have AhU = F, Ahu = F + T, and therefore AhE = T. For the hat discrete delta function, we have T = 0 for all i’s for the simple model. For the cosine discrete delta function | i| or other discrete delta functions, generally we have T = O(1/h) for a few grid points neighboring | j | the interface α, see Table 1 on page 19. So the interesting question is: Why is the IB method still first order accurate, that is, E = O(h)? To answer this question, we first introduce the following k k∞ lemma.

Lemma 2.1. Let Ahy = ek and y0 = yn = 0, where ek is the k-th unit base vector, then

hxi (1 xk) if i k, yi = − − ≤ (8) ( hxk (1 xi) otherwise. − −

The signiﬁcance of this lemma is that the solution is order h smaller than the concentrated source. Proof: We note the following identity e A y = h k = h δhat e . (9) h h h k

3 For this simple case, the IB method using the hat discrete delta function is identical to the IIM, see [13]. Thus from the Immersed Interface Method, see [3, 13], we know that y is the exact discrete solution at the grid points of the following boundary value problem

u00 = h δ(x x ), 0 < x < 1, u(0) = u(1) = 0, (10) − k whose solution is hx (1 xk) if x xk, y(x) = − − ≤ (11) ( hxk (1 x) otherwise. − − This completes the proof. Note that y(x) h. From this lemma, we have the following corollary. | | ≤

Corollary 2.2. Let Ahy = r, then y(x) h W r , where W is the number of non-zero compo- | | ≤ k k∞ nents of r.

The proof is straightforward from (11) and the fact that 0 x 1 and 0 1 x 1. Notice ≤ ≤ ≤ − ≤ that for a discrete delta function, it should satisfy at least the zeroth moment equation, see [3], that is δ (x α) = 1, (12) h i − i X corresponding to the continuous case δ (x α)dx = 1. Now we are ready to prove the main h − theorem. R Theorem 2.3. Let u(x) be the solution to (1) and U is the solution obtained from the immersed boundary method (3) using a discrete delta function δh(x) for (1). Then U is ﬁrst order accurate, that is

E C¯h, (13) k k∞ ≤ where C¯ is a constant.

Proof: We can decompose the local truncation error into two groups

T = Treg + Tirreg, (14)

where Treg = 0 corresponds to the local truncation errors at regular grid points where δh(xi k k∞ − α) = 0 and the true solution is piecewise linear. Note that, we have

n 1 − reg irreg irreg Ti = T + T = O + T . (15) i=1 X X X X On the other hand, we also have

u(xi 1) 2u(xi) + u(xi+1) hat − − = cδ (x α) (16) h2 h i −

4 since the ﬁnite diﬀerence method using the discrete delta function gives the exact solution at all the grid points. Thus we have

n 1 − u(xi 1) 2u(xi) + u(xi+1) T = − − cδ (x α) i h2 − h i − i=1 X (17) n 1 n 1 − − = cδhat(x α) cδ (x α) = 0, h i − − h i − i=1 i=1 X X irreg irreg from the zeroth moment equation (12). Thus we have i Ti = 0. We divide Ti into two irreg,+ groups, one with all positive Ti’s denoted as Ti ; the other one is all the negatives denoted as irreg, irreg,+ irreg, irreg,+ P irreg, Ti −. Since Ti + Ti − = 0, Ti and Ti − must have the same order of the magnitude O(1/h) although those index i are diﬀerent except that x α W h is true for all P P | i − | ≤ irregular grid points. Because the solution is linear with c, we have

1 1 irreg,+ irreg, E = Ah− T = Ah− T + T − . (18)

From the solution expression, we know that, assuming that x α, l ≤ 1 1 irreg,+ irreg, El = Ah− T = Ah− T + T − irreg,+ irreg, = x T (1 x ) + T −(1 x ) − l  i − i j − j  i j X X  

irreg,+ irreg, = x (1 α) T + T − + O(W h) − l −  i j  i j X X = O(W h),  

after we expand all xi’s and xj ’s at α and since all related xi and xj are within W h distance from the interface α. The proof for xl > α is similar except that we need to use the solution for x > α. This completes the proof.

3 Proof of the convergence of the IB method in 2D

The discussion for 2D problems is much more challenging since the interface is often a curve instead of a point. In [14], the author has proved the ﬁrst order accuracy of the IB method for the Stokes equations with a periodic boundary condition in 2D based on existing estimates between the discrete Green function and the continuous one in [8]. However, there are almost no theoretical proof on the IB method for elliptic interface problems or other PDES with general boundary conditions. We will prove that the result obtained from the IB for the elliptic interface problem with a Dirichlet boundary condition is indeed ﬁrst order accurate in this section.

5 1

0.9 Ω+

0.8

0.7

0.6

0.5 Ω− Γ

0.4

0.3

0.2

Ω=Ω+∪ Ω−∪ Γ 0.1 ∂ Ω 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 1: A diagram of a 2D elliptic interface problem. The interface is Γ.

Consider the following 2D elliptic interface problem,

∆u(x, y) = f(x, y) + v(s)δ (x X(s)) (y Y (s)) ds, (x, y) Ω, Γ − − ∈ (19) u(x, y) = u0(x, y),R |∂Ω where we assume that f C(Ω), Γ C1, v(s) C1. Without loss of generality, we assume that Ω ∈ ∈ ∈ is a unit square 0 x, y 1, see Fig. 1 for an illustration. The problem can be decomposed as the ≤ ≤ sum of the solutions of the following two problems. The ﬁrst one is

∆u1(x, y) = f(x, y) (20) u (x, y) = u (x, y), 1 |∂Ω 0 which is a regular problem whose solution u (x, y)] C2(Ω). The second problem is 1 ∈

∆u (x, y) = v(s)δ (x X(s)) (y Y (s)) ds, 2 − − ZΓ (21) u (x, y) = 0. 2 |∂Ω The solution to the second problem is equivalent to the following problem

∂u ∆u (x, y) = 0, [u ] = 0, 2 = v(s), 2 2 Γ ∂n Γ (22) u (x, y) = 0. 2 |∂Ω

6 The solution to the original problem is u = u1 + u2. Since u1 is the solution to a regular problem, it is enough just to consider u2(x, y). Thus we will simply use the notation u(x, y) for u2(x, y). Peskin’s IB method for the problem includes the following steps:

Generate a uniform Cartesian mesh x = ih, y = jh, i, j = 0, 1, , n. Here we use a uniform • i j · · · mesh for simplicity. We denote Ωh as the set of all grid points; and ∂Ωh as the grid points on the boundary.

Replace the partial derivatives with the ﬁnite diﬀerence approximation and use a discrete delta • function to spread the singular source to nearby grid points

Ui 1,j + Ui+1,j + Ui,j 1 + Ui,j+1 4Uij IB − − − = C , i, j = 1, 2, , n 1, h2 ij · · · − Nb (23) CIB = v δ (x X ) δ (y Y ) ∆s , ij k h i − k h j − k k Xk=1 where (X , Y ), k = 1, 2, , N , is a discretization of the interface Γ, and v v(s ), which k k · · · b k ≈ k we assume it is at least ﬁrst order approximation, vk = v(sk) + O(h).

Solve the ﬁnite diﬀerence system of equations above to get an approximation solution U . • { ij} This can be done by calling a fast Poisson solver, say [1].

3.1 Discrete delta functions and discrete Green functions

As a common practice, we assume that max s = ∆s O(h). In Peksin’s IB method, a discrete k{ k} ∼ delta function is used for two purposes. One is to spread the singular source to the nearby grids. The other one is to interpolate a grid function, say the velocity, to get its values on the interface. Thus the discrete delta function used should satisﬁes at least the zeroth moment condition as described in [3]. The interpolation using the discrete delta function should be at least ﬁrst order accurate, that is,

n 1 Nb − h2 v δ (x X ) δ (y Y ) ∆s = v(s)ds + O(h), (24) k h i − k h j − k k i,j=1 Γ X kX=1 Z which corresponds to

v(s)δ (x X(s)) δ (y Y (s)) dsdxdy = v(s)ds. (25) − − ZZΩ ZΓ ZΓ From u(x, y)δ (x X) δ (y Y ) dxdy = u(X, Y ), we should also have the interpolation property, Ω − − RR n 1 − h2u(x , y )δ (x X ) δ (y Y ) = u(X, Y ) + O(h). (26) i j h i − k h j − k i,j=1 X

7 In (24) and (26), the error terms depend on the ﬁrst order derivatives of v(s) and u(x, y), respectively. The discrete delta function has a compact support, that is,

δ (x X ) = 0, if x X > W h, and δ (y Y ) = 0, if y Y > W h, (27) h i − k | i − k| h j − k | j − k| where xij = (xi, yj ), and W is a constant. We deﬁne the error vector as E = E , where E = u(x , y ) U . The local truncation error { ij } ij i j − ij is deﬁned as T = T , { ij } u(xi 1, yj ) + u(xi+1), yj ) + u(xi, yj 1) + u(xi, yj+1) 4u(xi, yj) IB T = − − − C . (28) ij h2 − ij

In the matrix vector form, we have AhU = F, Ahu = F + T, and therefore AhE = T, where Ah is the matrix formed by the discrete Laplacian. We have T = O(h2) at regular grid points where | ij | CIB = 0. In general, we have T = O(1/h) for grid points neighboring the interface Γ except ij | ij | for the correction terms using the Immersed Interface Method (IIM) [11, 12, 13] for which we have T IIM = O(h) at irregular grid points where the interface cuts through the standard ﬁve-point | ij | stencil. It is interesting that the local truncation errors can have order O(1/h) at some grid points, but the global error is still of O(h). There has to be some kind of cancelations of the errors, which can be seen from our proof process.

Deﬁnition 3.1. Let elm be the unit grid function whose values are zero at all grid points except at xlm = (xl, ym) where its component is elm = 1. The discrete Green function centered at xlm with homogeneous boundary condition is deﬁned as

h 1 1 h G (x , x ) = (A )− e , G (∂Ω , x ) = 0, (29) ij lm h lm h2 h lm ij where ∂Ωh denotes the grid points on the boundary ∂Ω.

Note that from Remark 4.4.6 in [7], we know that Gh(xij , xlm) is symmetric,

h h G (xij , xlm) = G xlm, xij . (30) The usual discrete Green function, also called a discrete fundamen tal solution, on the entire integer lattice is defined as 1 , if xij = xlm, ∆ g (x , x ) = h2 (31) h h ij lm   0, otherwise, for all integers i and j, see for example, [2, 6, 7, 10, 14, 17, 18] for more discussions about the discrete Green’s functions. Note that gh (xij , xlm) is also symmetric, that is, gh (xij , xlm) = gh (xlm, xij ). To prepare the convergence proof, we first list some lemmas that are either directly or indirectly used in the convergence proof. The following lemma is the discrete first Green’s formula. Although it is not directly used in the proof, it shows how the discrete summation is related to the integral form; and how the source distribution is related to the jump conditions.

8 Lemma 3.2. The discrete ﬁrst Green’s formula and an error estimate. Let u(x, y) be the solution to (19). Thus u(x, y) is in the piecewise C 1(Ω) space, that is, u(x, y) ∈ C1(Ω Γ); Assuming that the distance between Γ and ∂Ω is O(1), that is, dist(Γ, ∂Ω) O(1), then \ ∼ we have

n 1 − ∂u ∆ u(x , y )h2 = ds + O(h) = v(s)ds + O(h), (32) h i j ∂n i,j=1 ∂Ω Γ X Z Z where

u(xi 1, yj ) + u(xi+1), yj ) + u(xi, yj 1) + u(xi, yj+1) 4u(xi, yj ) ∆ u(x , y ) = − − − , (33) h i j h2 is the discrete Laplacian using the standard ﬁve-point stencil, and the summation is over all the interior grid points.

Proof: We ﬁrst prove the discrete ﬁrst Green’s formula by expanding the summation. After cancelation of interior terms, only boundary terms are left in the summation as follows,

n 1 n 1 2 − u(x0, yj) u(x1, yj) − u(xn, yj ) u(xn 1, yj) ∆ u(x , y )h = h − + h − − h i j h h ij j=1 j=1 X X X n 1 n 1 − u(xi, y0) u(xi, y1) − u(xi, yn) u(xi, yn 1) + h − + h − − h h i=1 i=1 X X ∂u = ds + O(h). ∂n Z∂Ω On the other hand, by integrating both sides of the partial diﬀerential equation (19) with f(x, y) = 0 and u0(x, y) = 0, we get

∆udxdy = v(s)δ (x X(s)) (y Y (s)) ds dxdy, − − ZZΩ ZZΩ ZΓ or equivalently, ∂u ds = v(s) ds. ∂n Z∂Ω ZΓ This completes the proof.

Remark 3.3. The double integral Ω ∆udxdy can be divided into three parts RR ∆udxdy = ∆udxdy + ∆udxdy + ∆udxdy + − ZZΩ ZZΩ ZZΩ ZZΩ ∂u ∂u ∂u = ds ds + ds, ∂n − + ∂n + ∂n Z∂Ω ZΓ ZΓ

9 + Ω Ω Γ

Γ−

Ω− + Γ

Figure 2: A diagram of the domain, interface, and integration.

see Fig. 2 for an illustration, see also [12]. As 0, we have → ∂u lim ∆udxdy = v(s) ds = ds 0 ∂n → ZZΩ ZΓ ZΓ from the partial diﬀerential equation (19) with f(x, y) = 0 and u0(x, y) = 0. Thus we get ∂u lim ∆udxdy = ds. 0 ∂n → ZZΩ Z∂Ω While Lemma 3.2 is not directly used in our convergence proof, it is easier to illustrate the relation between the boundary integral along ∂Ω and the source distribution along Γ for the ﬁrst Green formula than the second Green formula that is used in the proof.

3.2 Interpolating the discrete delta function

h We know that G (xij , xlm) deﬁned in (29) is a grid function with the homogeneous Dirichlet bound- h ary condition at the boundary grid points. The discrete Laplacian ∆hG (xij , xlm) is zero at all 2 h interior grid points except that it is 1/h at xlm. We can interpolate G (xij , xlm) to the entire h domain to get an interpolation function GI (x, xlm). We consider any such an interpolation function that satisﬁes the following:

Gh(x , x ) = Gh(x , x ). • I ij lm ij lm Gh(x, x ) C1(Ω) H2(Ω). • I lm ∈ ∩ ∆ Gh(x , x ) = ∆ Gh(x , x ) = 0, that is, zero for all i and j except for i = l and j = m. • h I ij lm h ij lm The interpolation is second or higher order accurate, that is •

α h α h 3 α 1 ∂ G (x , x ) ∂ G (x , x ) Ch −k k , for α 2, (34) ij lm − I ij lm ≤ k k1 ≤

10 h where the derivatives of G (xij , xlm) is defined from finite differences, see [18], and α is the summation notation as used in the literature for the Sobolev spaces.

∆Gh(x, x )dxdy = O(h) except for the four neighboring squares centered at x on Rlm I lm lm • h which ∆G (x, xlm)dxdy = 1 + O(h). RR Rlm I RR We provide a construction of such an interpolation function in Appendix.

h h Lemma 3.4. Let GI (x, xlm) be an interpolation function of G (xij , xlm) that satisﬁes the conditions above, then we have the following estimates: 1 1 Gh (x, x ) + log x x 2 + h2 + O(h), (35) I lm ≤ 4 16 k − lmk2 α h C ∂ G (x, xlm ) + O(h), if α 1 k 1, (36) I α 1 ≤ ( xij xlm 2 + h)k k k k ≤ − k − k where k is the order of interpolation and C is a constant. The inequality (36) is true if dist(x, ∂Ω) ∼ O(1) and dist(∂Ω, x ) O(1). lm ∼ Proof: From Remark 4.4.8 in [7], we know that 1 1 Gh(x , x ) + log x x 2 + h2 . | ij lm | ≤ 4 16 k − lmk2 Thus from the order requirement of the interpolation function, we have the ﬁrst inequality. To prove the second inequality, we use the expression from (4.11) in [17]:

Gh (x , x ) = g (x , x ) sˆ(x , x ) s (x , x ) , (37) ij lm h ij lm − ij lm − ij lm h where sˆ(xij , xlm) is chosen such that ∆hG (xij , xlm) = 0 at the grid points on the boundary of the unit square using the method of images. Thus sˆ(xij , xlm) is a combination of gh(xij , xlm) α at points outside of the unit square. From Theorem 3.1 in [18], we have ∂ gh(xij , xlm) α 1 | | ≤ C/ ( xij xlm 2 + h)k k . The same can be said for sˆ(xij , xlm). The term s(xij , xlm) is chosen such k h − k that G (xij , xlm) = 0 at the grid points on the boundary of the unit square and ∆hs(xij , xlm) = 0 at all other grid points. Thus s(x , x ) is the trace of g (x , x ) sˆ(x , x ), see Remark ij lm |∂Ωh h ij lm − ij lm on page 297 in [5]. Since the values of s(x , x ) are from the discrete Laplacian, it can be ij lm |∂Ωh smoothly extended to the entire boundary ∂Ω. Thus from Lemma 2.7.5 in [9] and the maximum principle, we have

α 1 α 1 α C α 1 k k C α 1 k k ∂ s (x, xlm) k k sup s = k k sup s , (38) ≤ dist(x, ∂Ω) Ω | | dist(x, ∂Ω) ∂Ω | |

where C is another constan t. From Section 4.4 in [17], we know that s(xij , xlm) = O(1) as long as dist(∂Ω, x ) = O(1). Thus as long as dist(x, ∂Ω) O(1), we have ∂αs (x, x ) O(1). This lm ∼ lm ∼ completes the proof of the lemma.

11 Remark 3.5. The interpolation function is not unique. Along the boundary ∂Ω, from the require- h ment of the interpolation function, ∂GI (xij , xlm) /∂n does exist and continuous. We have, for example ∂Gh (x , x ) Gh (x , x ) Gh (x , x ) I ij lm = I 1j lm − I 0j lm + O(h) (39) ∂x h along the boundary x = 0.

By similar procedure in proving the discrete Green’s formula, we can get the second discrete Green’s formula.

h Lemma 3.6. Let u(x, y) be the solution to (19) and GI (x, xlm) be an interpolation function of Gh(x , x ) that satisﬁes the conditions listed in Section 3.2. If l, m = 1 or n 1, then we have, ij lm 6 − h 2 h ∆hu(xi, yj )G (xij , xlm)h = v(s)GI (X(s), xlm)ds + O(h). ij Γ X Z Proof: Again, we show the second discrete Green’s formula by expanding the summation. After cancelation of interior terms, only boundary terms and a source are left. Thus, we get

n 1 n 1 − − u(x , y ) u(x , y ) ∆ u(x , y )Gh(x , x )h2 = h 0 j − 1 j Gh(x , x ) h i j ij lm h 1j lm i,j=1 j=1 X X n 1 n 1 − − u(xn, yj) u(xn 1, yj ) h u(xi, y0) u(xi, y1) h + h − − G (xn 1,j , xlm) + h − G (xi1, xlm) h − h j=1 i=1 X X n 1 n 1 h h − u(xi, yn) u(xi, yn 1) h − G (x0j , xlm) G (x1j , xlm) + h − − G (xi,n 1, xlm) h − u(x1, yj ) h − − h i=1 j=1 X X n 1 h h n 1 h h − G (xn,j , xlm) G (xn 1,j , xlm) − G (xi0, xlm) G (xi1, xlm) h − − u(xn 1, yj ) h − u(xi, y1) − h − − h j=1 i=1 X X n 1 n 1 − h h − G (xi,n, xlm) G (xi,n 1, xlm) h 2 h − − u(xi, yn 1) + u(xi, yj )∆hG (xij , xlm)h − h − i=1 i,j=1 X X ∂u ∂Gh(x, x ) = (x) Gh(x, x ) I lm u(x) ds + u(x ) + O(h). ∂n I lm − ∂n lm Z∂Ω On the other hand, by integrating both sides of the partial diﬀerential equation (19) with f = 0 and u0 = 0, we get

Gh(x, x )∆udxdy = v(s)δ (x X(s)) (y Y (s)) ds Gh(x, x )dxdy, I lm − − I lm ZZΩ ZZΩ ZΓ or equivalently, ∂u ∂Gh(x, x ) Gh(x, x ) I lm u ds + u∆Gh(x, x )dxdy = v(s) Gh(X(s), x )ds. ∂n I lm − ∂n I lm I lm Z∂Ω ZZΩ ZΓ

12 Note that

h h h u(x)∆GI (x, xlm)dxdy = u(x)∆GI (x, xlm)dxdy + u(x)∆GI (x, xlm)dxdy Ω Ω Rlm ZZ Rij Rlm ZZ ZZ X\

h = u(xij ) + O(h) O(h)dxdy + u(xlm) + O(h) ∆GI (x, xlm) dxdy Rij Rlm XRij ZZ ZZ

3 h = u(xij )O(h ) + u(xlm) ∆GI (x, xlm)dxdy + O(h) Rlm XRij ZZ

= O(h) + u(xlm) + O(h) = u(xlm) + O(h), where Rij is the square centered at xij . This completes the proof.

IB 1 Lemma 3.7. Let Cij be the correction terms in the immersed boundary method (23), v(s) C h h ∈ be deﬁned in (19), and GI (x, xlm) be an interpolation function of G (xij , xlm) that satisﬁes the conditions listed in Section 3.2. Then we have the following estimate.

IB h 2 h Cij G (xij , xlm)h = v(s)GI (X(s), xlm)ds + O(h log h). ij Γ X Z

Proof: We denote h = max ∆s h. s { k} ∼

Nb CIBGh(x , x )h2 = v δ (x X ) δ (y Y ) ∆s Gh(x , x )h2 ij ij lm k h i − k h j − k k ij lm ij ij X X Xk=1 Nb = v ∆s δ (x x) δ (y y) Gh (x , x ) h2 k k h i − h j − I ij lm ij kX=1 X

Nb h = vk∆sk GI (Xk, xlm) + Ek . k=1 X From the expression (4.2) and Lemma 4.1 in [3], we know that

E Ch ∂αGh(ξ , x ) . | k| ≤ I k lm α 1=1 k Xk

Nb h The summation k=1 vk∆sk GI (Xk, xlm) + Ek is the composite trapezoidal rule for the line integral. We divide the summation into three groups, one with the summation of k that X P k k − x √h, one with h X x √h, and the other is for the rest of k’s. The contributions lmk2 ≥ ≤ k k − lmk2 ≤ from the boundary points are split as half and half for each group. For the ﬁrst two groups, we have

h h 1 00vk∆sk GI (Xk, xlm) + Ek = v(s)GI (X(s), xlm) ds + Eh, h X x Γ Γx k lm 2 h Z − lm k −X k ≥

13 where 00 means the coeﬃcients is half at the two boundary points. For the part X x √h, k k − lmk2 ≥ we have

1 2 1 2 1 Eh Chs max Chs Ch, ≤ k 2 ≤ 2 ≤ ( Xk xlm 2 + h) √h + h k − k h due to the second order partial derivatives of GI (X(s), xlm) and the ﬁrst order derivatives of Ek. For the part h X x √h, the error estimate is a little bit tricky. From the error ≤ k k − lmk2 ≤ estimate of the trapezoidal rule in each interval, we have

2 1 hs α h 2 hs Eh hs ∂ GI (ξk, xlm) Chs ≤ 12 ≤ x 2 k α 1=2 k ξk lm) 2 + h X k Xk X k − k √hs dr √hs dr Ch2 Ch2 Ch. ≤ s 2 ≤ s r2 ≤ Zhs r + h Zhs For the last group, we have

h h 2 00vk∆sk GI (Xk, xlm) + Ek = v(s)GI (X(s), xlm) ds + Eh h X x Γx k lm 2 h Z lm k −X k ≤ max v(s)h log h + E2, ≤ s h

h 2 from the estimate of GI (X(s), xlm) in (36). For the error term Eh, since the length of the integral is O(h), we have 1 1 E2 Ch2h max Ch3 Ch. h X 2 2 ≤ k ( Xk xlm 2 + h) ≤ (h + h) ≤ k − k

This completes the proof of the lemma. Now we are ready to prove the main result of the paper.

Theorem 3.8. Let u(x, y) be the solution to (19) and U is the solution obtained from the immersed boundary method (23) using a discrete delta function δh(x) for (19). Then U is ﬁrst order accurate with a logarithm factor in the L∞ norm, that is,

E Ch log h, i, j = 1, 2, , n 1. (40) | ij | ≤ · · · −

14 Proof: Consider the error at a grid point Elm, if xlm is close to the interface, that is, dist(Γ, x ) W h, we have lm ≤ 1TIB Elm = (Ah)− lm 1 IB 1 IB = (A )− T + (A )− T h reg lm h irr lm 2 1TIB = O(h ) + (Ah)− irr lm 1 2 1e 2 = h Tij (Ah)− ij 2 + O(h ) x h lm dis( ij ,Γ) W h X≤ = h2 ∆ u(x , y ) CIB Gh(x , x ) + O(h2) h i j − ij ij lm dis(xij ,Γ) W h X≤ = h2∆ u(x , y )Gh(x , x ) h2CIB Gh(x , x ) + O(h2) h i j ij lm − ij ij lm ij ij X X

h h = v(s)GI (X(s), xlm)ds vkGI (Xk), xlm)∆sk + O(h) Γ − ! Z Xk = O(h log h), after we apply Lemma 3.6 and Lemma 3.7. Note that, in the expansion of the summation from 2 dis(xij , Γ) W h to all interior grid points, we have used the fact that ∆hu(xi, yj ) = O(h ) and IB ≤ Cij = 0 when dis(xij , Γ) > W h. If dist(Γ, xlm) > W h, the proof above is still valid except that we are not going to have the singular integration. Thus, we do not need to have the log h factor. This means that for IB method, the larger errors often occur near or on the interface.

4 Conclusions and acknowledgments

We give a convergence proof of the immersed boundary (IB) method in the L∞ norm. The key of the proof is to establish a connection between the discrete Green function and a continuous one with the same boundary conditions. We show that the IB method is indeed ﬁrst order accurate with a log h factor if a reasonable discrete delta function is used. The conclusion should be applicable to other linear boundary conditions in addition to Dirichlet type as long as the method of images apply. The author was partially supported by the US ARO grants 550694-MA, the AFSOR grant FA9550-09-1-0520, the US NSF grant DMS-0911434, and the NIH grant 096195-01. The author is grateful for many valuable suggestions and comments from Dr. J. Thomas Beale of Duke University and Dr. Kazufumi Ito of North Carolina State University.

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A The construction of an interpolation function of the discrete Green function

h We construct an interpolation function from the grid function ∆GI (xij , xlm) to of the discrete Green h function ∆GI (x, xlm) that satisﬁes the conditions listed on page 10. Let φ(x, y) be the function that satisﬁes the following conditions:

• 1 ∆Gh(x, x ) = δ (x, y), δ (x, y)dxdy = . (41) I lm h h h2 ZZlm ∆Gh(x, x ) takes values of Gh(x , x ) at four corners and the center. • I lm I ij lm In other words, we choose the source term and the boundary condition so that the above two conditions are satisﬁed.

h Then, we construct the interpolation function GI (x, xlm) on other squares from the values of ∆Gh(x, x ) using Q (K) C1(Ω), where Q is defined as the finite element space I lm k ∈ k k Q (K) = v(x, y), on each R : v(x, y) = β xiyj , v(x, y) C1(Ω) (42) k  îˆj ij ∈  i=0,j=0  X  over a quadrilateral mesh K, Rîˆj is a square of h by h. In the paper [21], the authorhas proposed 1 a system way to construct C -Qk finite element spaces on quadrilaterals meshes. We have implemented and tested the interpolation function Gh(x, x ) of Q (K) C1(Ω), which I lm 5 ∈ is third order accurate (k = 3). As usual, we just need to construct a shape function over a unit square. The total degree of freedom of Q (K) C1(Ω) is 36. In our construction, we specify the 5 ∈ values of v, vx, vy, vxx, vxy, vyy, which imposes 24 constraints. To keep the continuity of the solution and first order partial derivatives, for example, along the side x = 0, we impose the coefficients of xy5 and xy4 to be zero. That is, there are two constraints along each side, which adds additional 8

17 constraints. Using the undetermined coefficient method for the 36 constraint with the 32 conditions, we have a system of equations with 36 unknowns and 32 equations (constraints). The condition number of the coefficient matrix, the ration of the largest and smallest non-zero singular values is 8.9851 103, which indicates the matrix has full row rank and the system of equations has infinite × number of solutions. We suggest to choose the SVD solution as the interpolation function. To see that the interpolation function is in C1, we use the side x = 0 as an example. Along this side, the function is a fifth polynomial, which is uniquely determined by its values, the first and second order derivatives at two points. Thus the interpolation function and its tangential derivative are continuous along this side. The normal derivative is in general still a fifth polynomial of y along x = 0. After we impose the constraints that its coefficients of xy5 and xy4 to be zero, it becomes a cubic polynomial which is uniquely determined by its values (vx) and tangential derivatives (vxy) at two points. In figure 3, we show a plot of such an interpolation function in two neighboring square with function value at one corner be a unit while others and all the derivatives are zero.

1.2

0.8

0.6

0.4

0.2

−0.2 1

0.8

0.6

0.4

0.2 1 0.6 0.8 0.2 0.4 −0.2 0 0 −0.6 −0.4 −1 −0.8

Figure 3: A plot of the shape function in Q (K) C1(Ω) at two neighboring squares. 5 ∈

B A numerical example

We use a numerical example to show that the immersed boundary method is indeed ﬁrst order accurate and its local truncation error is of order O(1/h). The diﬀerential equation is

∆u(x, y) = 2δ(x X(s))δ(y Y (s))ds, 1 < x, y < 1 (43) − − − ZΓ

18 where the interface Γ is the circle r = 1/2, where r = x2 + y2. The function

1 p if r 1 u(x, y) = ≤ 2 (44) 1 ( 1 + log(2r) if r > 2 satisﬁes the PDE. This example is taken from [11]. We use the IB method to solve the PDE with the Dirichlet boundary condition given by (44). In Table 1, we list the results of a grid reﬁnement analysis. In Table 1, the number N is the number of grid lines in the x and y directions. The interface r = 1/2 is discretized by X = 0.5 cos(2kπ/N) Y = 0.5 sin(2kπ/N), k = 0, 1, , N 1. k k · · · −

N Ecos ordercos T cos Ehat orderhat T hat ∞ ∞ ∞ ∞ k k 2 k k k k 2 k k 20 5.7217 10− 17.3534 2.1724 10− 9.1327 2 3 40 2.7226 10− 1.0715 34.9848 9.9933 10− 1.1202 20.6630 2 3 80 1.3399 10− 1.0228 69.9772 5.2761 10− 0.9215 43.5721 3 3 160 6.7340 10− 0.9926 139.254 4.5365 10− 2.1789 117.289 3 3 320 3.3510 10− 1.0066 279.483 1.8853 10− 1.2667 195.664 3 3 640 1.6737 10− 1.0018 556.360 1.1985 10− 0.6536 495.791 4 4 1280 8.4663 10− 0.88326 1122.01 5.4021 10− 1.1497 892.463

Table 1: A grid reﬁnement analysis of the IB method using discrete cosine and hat delta functions.

In the second and ﬁfth columns of Table 1, we show the L∞ errors of the computed solution of the IB method using the discrete cosine and hat delta functions, respectively. The third and sixth columns, we show the approximate convergence order using the two consecutive errors. We see clearly ﬁrst order convergence. In the fourth and seventh columns, we show the local truncation errors in the L∞ norm. We see that the local truncation error increase in the order of O(1/h).